Double Percentage Calculator
Apply two percentage changes in sequence and see the combined change, the error from simply adding them, and the change needed to undo both.
Quick Presets
Updated Formula covered by automated tests
When one percentage change is applied on top of another, the second one acts on a number that has already moved. That is why a 20 percent markdown followed by an extra 10 percent off at the register is not 30 percent off, and why a stock that falls 50 percent has to double, not rise 50 percent, to get back to where it started. This calculator applies two changes in sequence, shows the value after each step, and reports the single equivalent change.
It also lines that true combined figure up against the naive answer you get by adding the two percentages, so you can see exactly how far off the shortcut is. Shoppers use it to check stacked coupons, employees to see what two raises in a row are worth, and investors to understand why a gain followed by an equal loss leaves them behind. The last result tells you what change would be needed to undo both steps and return to the starting value.
The formula
- start
- The value before any change, in any unit (dollars, units sold, points)
- p₁, p₂
- The first and second percentage changes; positive for an increase, negative for a decrease
- combined %
- The single percentage change that has the same overall effect as the two in sequence
- recovery %
- The change that would bring the final value back to the start
Each percentage change is really a multiplication by a growth factor: +25 percent multiplies by 1.25 and −20 percent multiplies by 0.80. Applying two changes means multiplying the two factors, and multiplication of factors is what makes the result differ from addition. Expanding the product gives p₁ + p₂ plus a cross term p₁ × p₂ ÷ 100. That cross term is the whole gap between the shortcut and the truth: it is positive when both changes have the same sign and negative when they have opposite signs.
Because multiplication is commutative, the order of the two changes never affects the final value, only the intermediate one. The same idea extends to any number of steps, which is how compound interest, inflation over several years and chained retail markdowns are all calculated.
Worked example
Marco puts $10,000 into a fund. It gains 25% in the first year and loses 20% in the second, and he wonders why his statement shows no profit when 25 − 20 suggests he should be up 5%.
- 1 After year one: 10,000 × 1.25 = $12,500.
- 2 After year two: 12,500 × 0.80 = $10,000.
- 3 Combined change: (1.25 × 0.80 − 1) × 100 = (1.00 − 1) × 100 = 0%.
- 4 Cross-term check: 25 + (−20) + (25 × −20 ÷ 100) = 5 − 5 = 0%.
- 5 The simple sum, +5%, overstates the true result by 5 percentage points.
Result: Marco ends exactly where he began, at $10,000. The 20% loss was taken from a larger balance than the one the 25% gain was earned on, which cancels the gain completely.
Before you rely on the result
- Only two sequential changes are entered here. For three or more, multiply all the growth factors, or apply the result of this page as the first change of another calculation.
- Investment returns quoted as averages are often arithmetic means; the compounded result shown here is what actually happens to the balance, and fees or contributions between periods are not included.
Frequently asked questions
Is 20% off plus an extra 10% off the same as 30% off?
No. The extra 10% is taken from the already reduced price, so the multiplier is 0.80 × 0.90 = 0.72, which is 28% off. On a $150 jacket that is $108 instead of $105, a $3 difference that grows with the price.
Why does a 50% loss need a 100% gain to recover?
After a 50% fall, the value is half of what it was, so it has to double to get back. In general the recovery needed after a fall of d percent is d ÷ (100 − d) × 100 percent, which climbs steeply: a 10% fall needs 11.1%, a 33.3% fall needs 50%, and a 75% fall needs 300%.
Does it matter which percentage is applied first?
Not for the final figure, since the two growth factors are multiplied and multiplication can be done in either order. Only the value after the first step changes. Some stores do apply percentage coupons and fixed-dollar coupons in a particular order, and mixing those two kinds of discount does change the result.
What are two 3% raises worth compared with one 6% raise?
Slightly more: 1.03 × 1.03 = 1.0609, a 6.09% total increase. On a $60,000 salary that is $63,654 instead of $63,600. The cross term (3 × 3 ÷ 100 = 0.09 points) is small for small percentages and large for big ones.